a b s t r a c tBy relying on the definition of admissible boundary conditions, the principle of virtual work and some kinematical considerations, we establish the skew-symmetric character of the couple-stress tensor in size-dependent continuum representations of matter. This fundamental result, which is independent of the material behavior, resolves all difficulties in developing a consistent couple stress theory. We then develop the corresponding size-dependent theory of small deformations in elastic bodies, including the energy and constitutive relations, displacement formulations, the uniqueness theorem for the corresponding boundary value problem and the reciprocal theorem for linear elasticity theory. Next, we consider the more restrictive case of isotropic materials and present general solutions for two-dimensional problems based on stress functions and for problems of anti-plane deformation. Finally, we examine several boundary value problems within this consistent size-dependent theory of elasticity.

In this paper, a consistent theory is developed for size-dependent piezoelectricity in dielectric solids. This theory shows that electric polarization can be generated as the result of coupling to the mean curvature tensor, unlike previous flexoelectric theories that postulate such couplings with other forms of curvature and more general strain gradient terms ignoring the possible couple-stresses. The present formulation represents an extension of recent work that establishes a consistent size-dependent theory for solid mechanics. Here by including scale-dependent measures in the energy equation, the general expressions for force-and couple-stresses, as well as electric displacement, are obtained.Next, the constitutive relations, displacement formulations, the uniqueness theorem and the reciprocal theorem for the corresponding linear small deformation size-dependent piezoelectricity are developed. As with existing flexoelectric formulations, one finds that the piezoelectric effect can also exist in isotropic materials, although in the present theory the coupling is strictly through the skew-symmetric mean curvature tensor.In the last portion of the paper, this isotropic case is considered in detail by developing the corresponding boundary value problem for two dimensional analyses and obtaining a closed form solution for an isotropic dielectric cylinder.

Fundamental solutions for two-and three-dimensional linear isotropic size-dependent couple stress elasticity are derived, based upon the decomposition of displacement fields into dilatational and solenoidal components. While several fundamental solutions have appeared previously in the literature, the present version is for the newly developed fully determinate couple stress theory. Within this theory, the couple stress tensor is skewsymmetrical and thus possesses vectorial character. The present derivation provides solutions for infinite domains of elastic materials under the influence of unit concentrated forces and couples. Unlike all previous work, unique solutions for displacements, rotations, force-stresses and couple-stresses are established, along with the corresponding force-tractions and couple-tractions. These fundamental solutions are central in analysis methods based on Green's functions for infinite domains and are required as kernels in the corresponding boundary integral formulations for size-dependent couple stress elastic materials.

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SUMMARYCouple-stresses are introduced to account for the microstructure of a material within the framework of continuum mechanics. Linear isotropic versions of such materials possess a characteristic material length l that becomes increasingly important as problem dimensions shrink to that level (e.g., as the radius a of a critical hole reduces to a size comparable to l/. Consequently, this size-dependent elastic theory is essential to understand the behavior at micro-and nano-scales and to bridge the atomistic and classical continuum theories. Here we develop an integral representation for two-dimensional boundary value problems in the newly established fully determinate theory of isotropic couple stress elastic media. The resulting boundary-only formulation involves displacements, rotations, force-tractions and moment-tractions as primary variables. Details on the corresponding numerical implementation within a boundary element method are then provided, with emphasis on kernel singularities and numerical quadrature. Afterwards the new formulation is applied to several computational examples to validate the approach and to explore the consequences of size-dependent couple stress elasticity.